appendix b: equation list - nikola engineering · · 1999-06-20(δ1=0.001) b.1.2 effective...

BSIM3v3 Manual Copyright © 1995, 1996, UC Berkeley B-1

APPENDIX B: Equation List

B.1 I-V Model

B.1.1 Threshold Voltage

V V K V K V

KN

LK K V

T

W W

D DW L

lD

W L

lV

D DL

lD

L

lV

DL

lD

L

l

th tho s bseff s bseff

LX

effs b bseff

OX

effs

VT w VT weff eff

twVT w

eff eff

twbi s

VT VTeff

tVT

eff

tbi s

subeff

tosub

eff

to

= + − − −

+ + −

+

+

− − + −

−

− − + −

−

− − + −

+

1 2

1 3 30

0 1 1

0 1 1

1 1

22

22

22

( )

( )

exp( ) exp( ) ( )

exp( ) exp( ) ( )

exp( ) exp( )

'

' '

Φ Φ

Φ Φ

Φ

Φ

+( )E E V Vtao tab bseff ds

l X C D Vt si dep ox VT bseff= +ε / ( )1 2

l X C D Vtw si dep ox VT w bseff= +ε / ( )1 2

l X Cto si dep ox= ε 0 /

XV

qNdep

si s bseff

ch= −2ε ( )Φ

I-V Model

B-2 BSIM3v3 Manual Copyright © 1995, 1996, UC Berkeley

(δ1=0.001)

B.1.2 Effective Vgs-Vth

B.1.3 Mobility

For Mobmod=1

XqN

depsi s

ch0

2= ε Φ

V V V V V V Vbseff bc bs bc bs bc bc= + − − + − − −0 5 41 12

1. [ ( ) ]δ δ δ

V sK

Kbc = −0 9

1

4 2

2

2. ( )φ

V vN N

nbi t

ch DS

i= ln( )

2

V

n vV V

n v

n Cq N

V V V

n v

gsteff

tgs th

t

OXs

si ch

gs th off

t

=+

−

+ −− −

2 12

1 22 2

2

ln exp( )

exp( )Φ

ε

n NC

C

C C V C V DL

lD

L

lC

C

Cfactor

d

ox

dsc dscd ds dscb bseff VTeff

tVT

eff

t

ox

it

ox= + +

+ + − + −

+1 221 1( ) exp( ) exp( )

CX

dsi

dep= ε

I-V Model


For Mobmod=2

For Mobmod=3

B.1.4 Drain Saturation Voltage

For Rds>0 or λ≠1:

µ µeff

o

a c bseffgsteff th

OXb

gsteff th

OXU U V

V V

TU

V V

T

=+ + + + +

12 2 2( )( ) ( )

µ µeff

o

a c bseffgsteff

OXb

gsteff

OXU U V

V

TU

V

T

=+ ++1 2( )( ) ( )

µ µeff

o

agsteff th

OXb

gsteff th

OXc bseffU

V V

TU

V V

TU V

=+ + + + +1

2 212[ ( ) ( ) ]( )

Vb b ac

adsat = − − −2 4

2

a A W C R Abulk eff sat ox DS bulk= + −2 11ν

λ( )

I-V Model


For Rds=0, λ=1:

B.1.5 Effective Vds

b V v A E L A V v W C Rgsteff t bulk sat eff bulk gsteff t eff sat ox DS= − + − + + +

( )( ) ( )22

1 3 2λ

ν

c V v E L V v W C Rgsteff t sat eff gsteff t eff sat ox DS= + + +( ) ( )2 2 2 2 ν

λ = +A V Agsteff1 2

VE L V v

A E L V vdsat

sat eff gsteff t

bulk sat eff gsteff t= +

+ +( )

( )

2

2

AK

V

A L

L X XA V

L

L X X

B

Weff B K Vbulk

s bseff

eff

eff J dep

gs gsteffeff

eff J dep

o

ETA bseff= +

− +−

++

+ +( { [ ( ) ]

'})1

2 21

2

1

1

1 0 2

1Φ

Esatsat

eff= 2ν

µ

( )V V V V V V Vdseff dsat dsat ds dsat ds dsat= − − − + − − +1

242δ δ δ( )

I-V Model


B.1.6 Drain Current Expression

IIR I

V

V V

V

V V

Vds

dso Vdseff

ds dso Vdseff

dseff

ds dseff

A

ds dseff

ASCBE=

++ −

+ −

( )

( )1

1 1

IW C V A

V

V vV

L V E Ldso

eff eff ox gsteff bulkdseff

gsteff tdseff

eff dseff sat eff=

−+

+

µ (( )

)

[ / ( )]

12 2

1

V VP V

E L V VA Asat

vag gsteff

sat eff ACLM ADIBLC= + + + −( )( )1

1 1 1

VA E L V

P A E litlV VACLM

bulk sat eff gsteff

CLM bulk satds dseff= + −( )

VV v

t P V

A V

A V V vADIBLC

gsteff t

rou DIBLCB bseff

bulk dsat

bulk dsat gsteff t= +

+−

+ +

( )

( )

2

11

2θ

θrout DIBLC ROUTeff

tROUT

eff

tDIBLCP D

L

lD

L

lP= − + −

+10 0

22

2exp( ) exp( )

1 2 1

V

P

L

P litl

V VASCBE

scbe

eff

scbe

ds dseff=

−−

exp

I-V Model


B.1.7 Substrate Current

B.1.8 Polysilicon Depletion Effect

VE L V R C W V

A V

V vR C W A

Asat

sat eff dsat DS sat ox eff gsteffbulk dsat

gsteff t

DS sat ox eff bulk=

+ + −+

− +

2 12 2

2 1

ν

λ ν

[( )

]

/

litlT Xsi ox j

ox=

εε

IL

V VV V

IR I

V

V V

Vsub

o

effds dseff

o

ds dseff

dso

ds dso

dseff

ds dseff

A= − −

− ++ −

α β( ) exp( )

11

V X EqN X

poly poly polypoly poly

si= =1

2 2

2

ε

Ngate

ε ε εox ox si poly si poly polyE E q N V= = 2 Ngate

V V V Vgs FB s poly ox− − = +φ

a V V V Vgs FB s poly poly( )− − − − =φ 2 0

I-V Model


B.1.9 Effective Channel Length and Width

aε2

ox

2qεsiNgateT2ox

---------------------------------------=

V Vq N T V V

q N Tgs eff FB s

si g ox

ox

ox gs FB s

si poly ox_ (

( ))= + + +

− −−φ

ε

ε

ε φ

ε

2

2

2

21

21

Ngate

qεsiNgateTox2

L L dLeff drawn= − 2

W W dWeff drawn= − 2

W W dWeff drawn' '= − 2

dW dW dW V dW Vg gsteff b s bseff s= + + − −' ( )φ φ

dW WW

L

W

W

W

L Wl

Ww

Wwnwl

W Wwn'

int ln ln= + + +

dL LL

L

L

W

L

L Wl

LwLwn

wlL Lwn= + + +int ln ln

I-V Model


B.1.10Drain/Source Resistance

B.1.11Temperature Effects

RR V V

Wds

dsw wg gsteff wb s bseff s

effWr=

+ + − −[ P P ( )]

( )

r r

'

1

106

φ φ

V V K K L K V T Tth T th Tnorm T t l eff T bseff norm( ) ( ) ( / )( / )= + + + −1 1 2 1

µ µ µo T o Tnorm

T

Tnorm

te( ) ( )( )=

v v A T Tsat T sat Tnorm T norm( ) ( ) ( / )= − −1

R R TT

Tdsw T dsw norm t

norm( ) ( ) rP ( )= + −1

U U U T Ta T a Tnorm a norm( ) ( ) ( / )= + −1 1

U U U T Tb T b Tnorm b norm( ) ( ) ( / )= + −1 1

U U U T Tc T c Tnorm c norm( ) ( ) ( / )= + −1 1

Capacitance Model Equations


B.2 Capacitance Model Equations

B.2.1 Dimension Dependence

B.2.2 Overlap Capacitance (for NMOS)

B.2.2.1 Source Overlap Capacitance

(1) for capmod=0

(2) for capmod=1

δWL W L Weff = + + +DWCWl Ww Wwl

Wln Wwn Wln Wwn

δLL W L Weff = + + +DLCLl Lw LwlLln Lwn Lln Lwn

L L Lactive drawn eff= − 2δ

W W Wactive drawn eff= − 2δ

Q

WVoverlap s

activegs

, = CGS0



(3) for capmod=2

B.2.2.2 Drain Overlap Capacitance

(1) for capmod=0

if (Vgs <0)

Q

WV

Voverlap s

activegs

gs, = + − + −

CGS0

C C

CKAPPA

KAPPA GS1

21 1

4

else

Q

Woverlap s

active

, (= +CGS0 C C ) VKAPPA GS1 gs

( ) ( )V V V wheregs overlap gs gs, .= − + − −

=1

24 0 021 1

2

1 1δ δ δ δ

Q

WV V V

Voverlap s

activegs gs gs overlap

gs overlap,,

,= + − − − + +

CGS0 CGS1

CKAPPA

CKAPPA21 1

4+



(2) for capmod=1

(3) for capmod=2

Q

WVoverlap d

activegd

, = CGD0

if (Vgd <0)

Q

WV

Voverlap d

activegd

gd, = + − + −

CGD0

C C

CKAPPA

KAPPA GD1

21 1

4

else

Q

Woverlap d

active

, (= +CGD0 C C ) VKAPPA GD1 gd

( ) ( )V V V wheregd overlap gd gd, .= − + − −

=1

24 0 022 2

2

22δ δ δ δ

Q

WV V V

Voverlap d

activegd gd gd overlap

gd overlap,,

,= + − − − + +

CGD0 CGD1

CKAPPA

CKAPPA21 1

4+



B.2.2.3 Gate Overlap Charge

B.2.3 Instrinsic Charges

(1) for capmod=0

( )Q Q Qoverlap overlap overlap d,g ,s ,= − +

a) Accumulation region (Vgs <Vfb+Vbs)

( )Q W L C V V Vg active active ox gs bs fb= − −

Q Qsub g= −

Qinv = 0

b) Subthreshold region (Vgs <Vth)

( )Q W L C

V V Vb active active ox

gs fb bs= − − + +− −

K

K1

1

2

221 1

4



Q Qg b= −

Qinv = 0

c) Strong inversion (Vgs>Vth)

VV V

Adsat cvgs th

bulk, '

= −

A ALbulk bulk

eff

' = +

0 1

CLCCLE

AK

V

A L

L X X

B

W B K Vbulk

s bs

eff

eff J dep

o

eff ETA bs0

1 0

11

2 2

1

1= +

− ++

+

+Φ{ }

'

V V K Vth fb s s bsf= + −+Φ Φ1



otherwise

(i) 50/50 Charge partition

if Vds<Vdsat

Q C W L V VV A V

V VA V

g ox active active gs fb sds bulk ds

gs thbulk ds

= − − − +− −

[

( )

]'

'Φ2

122

2

Q W L C V VA V A V

V VA

Vinv active active ox gs th

bulk ds bulk ds

gs thbulk

ds

= − − − +− −

[' '

('

)]

2 122

2 2

Q W L C V VA V A A V

V VA

Vb active active ox fb th s

bulk ds bulk bulk ds

gs thbulk

ds

= − + +−

−−

− −[

( ' ) ( ' ) '

('

)]Φ

1

2

1

122

2

Q Q W L C V VA V A V

V VA

Vs d inv active active ox gs th

bulk ds bulk ds

gs thbulk

ds

= = = − − − +− −

0 5Q2 12

2

2 2

. [' '

('

)]

Q W L C V VV

g active active ox gs fb sdsat= − − −( )Φ3



(ii) 40/60 channel-charge Partition

if (Vds <Vdsat)

Q Q W L C V Vs d active active ox gs th= = − −1

3( )

Q W L C V VA V

b active active ox fb s thbulk dsat

= − + − +−

(( ' )

)Φ1

3

Q C W L V VV A V

V VA V


gs thbulk ds

= − − − +− −

[

( )

]'

'Φ2

122

2

Q W L C V VA V A V

V VA


bulk ds bulk ds

gs thbulk

ds

= − − − +− −

[' '

('

)]

2 122

2 2


V VA



gs thbulk

ds

= − + +−

−−

− −[

( ' ) ( ' ) '

('

)]Φ

1

2

1

122

2



otherwise

(iii) 0/100 Channel-charge Partition

if Vds <Vdsat

Q W L C

V V AV

A VV V A V V V A V

V VA

V

d active active ox

gs th bulkds

bulk dsgs th bulk ds gs th bulk ds

gs thbulk

ds

= −

− − +

− − − +

− −

2 26 8 40

2

2 2

2

' ' [( ) ' ( ) ( ' )

('

)

Q Q Q Qs g b d= − + +( )

Q W L C V VV


Q W L C V Vd active active ox gs th= − −4

15( )

Q Q Q Qs g b d= − + +( )

Q W L C V VA V


= − + − +−

(( ' )

)Φ1

3



otherwise

Q C W L V VV A V

V VA V


gs thbulk ds

= − − − +− −

[

( )

]'

'Φ2

122

2

Q W L C V VA V A V

V VA


bulk ds bulk ds

gs thbulk

ds

= − − − +− −

[' '

('

)]

2 122

2 2


V VA



gs thbulk

ds

= − + +−

−−

− −[

( ' ) ( ' ) '

('

)]Φ 1

2

1

122

2

Q W L CV V A

VA V

V VA

Vd active active ox

gs th bulkds

bulk ds

gs thbulk

ds

= − − +− −

−2 4 24

2

2' ( ' )

('

)

Q Q Q Qs g b d= − + +( )

Q W L C V VV




(2) for capmod=1

if (Vgs <Vfb+Vbs+Vgsteffcv)

else

Q W L C V VA V


= − + − +−

(( ' )

)Φ1

3

Qd = 0

Q Q Qs g b= − +( )

( )Q W L C V Vfb V Vg active active ox gs bs gsteffcv1 = − − − −

( )Q W L C

V V V cv Vg active active ox

gs FB gsteff bs1

2

221 1

4= − + +

− − −

K

K1

1

Q Qb g1 1= −



if (Vds <=Vdsat)

VV

Adsat cvgsteffcv

bulk, '

=

A ALbulk bulk

eff

'= +

0 1

CLCCLE

AK

V

A L

L X X

B

W B K Vbulk

s bseff

eff

eff J dep

o

eff ETA bseff0

1 0

11

2 2

1

1= +

− ++

+

+Φ{ }

'

V n vV V

n vgsteffcv t

gs th

t= + −

ln exp( )1

Q Q W L C VV A V

VA

Vg g active active ox gsteffcv

ds bulk ds

gsteffcvbulk

ds

= + − +−

1

2

212

2

''

( )Q Q W L C

AV

A A V

VA

Vb b active active ox

bulkds

bulk bulk ds

gsteffcvbulk

ds

= +−

−−

−

1

21

2

1

122

' ' '

'



(i) 50/50 Channel-charge Partition

(ii) 40/60 Channel-charge partition


Q QW L C

VA

VA V

VA

Vs d

active active oxgsteffcv

bulkds

bulk ds

gsteffcvbulk

ds

= = − − +−

2 212

2

2 2' ''

( ) ( ) ( )

QW L C

VA

V

V V A V V A V A V

sactive active ox

gsteffcvbulk

ds

gsteffcv gstefcvf bulk ds gsteffcv bulk ds bulk ds

= −−

− + −

22

4

3

2

3

2

15

2

3 2 2 3

'

' ' '

Q Q Q Qd g b s= − + +( )

( )Q W L C

V A V A V

VA

Vs active active ox

gstefcv bulk ds bulk ds

gsteffcvbulk

ds

= − + −−

2 424

2

2' '

'

Q Q Q Qd g b s= − + +( )



if (Vds >Vdsat)

(i) 50/50 Channel-charge Partition

(ii) 40/60 Channel-charge Partition


Q Q W L C VV

g g active active ox gsteffcvdsat= + −

1

3

( )Q Q W L C

V Vb b active active ox

gsteffcv dsat=−

−13

Q QW L C

Vs dactive active ox

gsteffcv= = −3

QW L C

Vsactive active ox

gsteffcv= −2

5

Q Q Q Qd g b s= − + +( )

Q W L CV

s active active oxgstefcv= −

2

3



(3) for capmod=2

Q Q Q Qd g b s= − + +( )

( )Q Q Q Q Qg inv acc sub sub= − + + +0 δ

Q Q Q Qb acc sub sub= + +0 δ

Q Q Qinv s d= +

{ }V vfb V V vfb where V vfb VFBeff gb= − + + = − − =0 5 4 0 023 32

3 3 3 3. ; .δ δ δ

vfb th s sV K= − −φ φ1

( )Q W L C V vfbacc active active ox FBeff= − −

( )Q W L C

V V V cv Vsub active active ox

gs FBeff gsteff bs

0

2

221 1

4= − − + +

− − −

gamma

gamma1

1

K12

K12

cv - Vbseff

VV

Adsat cvgstefcvf

bulk, '

=gsteff,cv



B.2.3.1 50/50 Charge partition

A ALbulk bulk

active

' = +

0 1

CLCCLE

AK

V

A L

L X X

B

W B K Vbulk

s bseff

eff

eff J dep

o

eff ETA bseff0

1 0

11

2 2

1

1= +

− ++

+

+Φ{ }

'

V n vV V

n vgsteffcv t

gs th

t= + −

ln exp( )1

{ }V V V V V where V V Vcveff dsat cv dsat cv dsat cv ds= − + + = − − =, , ,. ; .0 5 4 0 024 42

4 4 4 4δ δ δ

Q W L C VA

VA V

VA

Vinv active active ox gsteffcv

bulkcveff

bulk cveff

gsteffcvbulk

cveff

= − −

+

−

' '

'212

2

2 2

( )δQ W L C

AV

A A V

VA

Vsub active active ox

bulkcveff

bulk bulk cveff

gsteffcvbulk

cveff

=−

−−

−

1

2

1

122

2' ' '

'



B.2.3.2 40/60 Channel-charge Partition

B.2.3.3 0/100 Charge Partition

Q Q QW L C

VA

VA V

VA

Vs d inv

active active oxgsteffcv

bulkcveff

bulk cveff

gsteffcvbulk

cveff

= = = − − +−

0 52 2

122

2 2

.' '

( ) ( ) ( )QW L C

VA

V

V V A V V A V A Vsactive active ox

gsteffcvbulk

cveff

gsteffcv gstefcvf bulk cveff gsteff bulk cveff bulk cveff=−−

− + −

22

4

3

2

3

2

1523 2 2 3

'' ' 'gsteffcv

( ) ( ) ( )QW L C

VA

V

V V A V V A V A Vdactive active ox

gsteffcvbulk

cveff

gsteffcv gsteffcv bulk cveff gsteffcv bulk cveff bulk cveff= −−

− + −

22

5

3

1

523 2 2 3

'' ' 'gsteffcv

( )Q W L C

V A V A V

VA

Vs active active ox

gstefcvf bulk cveff bulk cveff

gsteffcvbulk

cveff

= − + −−

2 424

2

2' '

'gsteffcv

( )Q W L C

V A V A V

VA

Vd active active ox

gsteffcv bulk cveff bulk cveff

gsteffcvbulk

cveff

= − − +−

2

3

48

2

2' '

'

NQS Model Equations:


B.2.4 Intrinsic Capacitances (with Body bias and DIBL)

B.3 NQS Model Equations:

Quasi-static equilibrium channel charge:

Actual channel charge and Qdef obtained from subcircuit (Figure 5-2):

( )CQ

V

V

Vs d g b g

s d g b

gsteffcv

gsteffcv

gt, , , ,

, , ,=∂∂

∂∂

( )CQ

V

Q

V

V

V

V

V

V

Vs d g b s

s d g b

ds

s d g b

gsteffcv

gsteffcv

gt

th

ds

th

bs, , , ,

, , , , , ,= − + +

∂∂

∂∂

∂∂

∂∂

∂∂

( )CQ

V

Q

V cv

V

V

V

Vs d g b d

s d g b

ds

s d g b

gsteff

gsteffcv

gt

th

ds, , , ,

, , , , , ,= −∂

∂∂∂

∂∂

∂∂

( )CQ

V

Q

V

V

V

V

Vs d g b b

s d g b

bs

s d g b

gsteffcv

gsteffcv

gt

th

bs, , , ,

, , , , , ,= −∂

∂∂∂

∂∂

∂∂

Q Q Qeq g b= − +( )

Q Q Qch eq def= −

gdrift diff

τ τ τ τ= = +1 1 1

Flicker Noise


where,

and

B.4 Flicker Noise

There exists two models for flicker noise. Each of these can be toggled by the

noimod flag.

1. For noimod=1 and 4


1. Vgs>Vth+0.1:

τµ ε α

driftox eff eff

eff eq def

C W L

Q Q=

−

3

≈ ζQeq

ε ≡ =Elmore Constant ( )default 5 0 0 10 05. . ( . )≤ ≤ =α default

ζµ ε

=C W Lox eff eff

eff

3

τµdiff

eff

eff

qL

KT=

2

16

Flic NoiseK I

C L f

f dsaf

ox effefker = 2

Flic Noisevtq I

f L CN

No x

Nl xN No Nl

N No NlvtI L

f L W

N N Nl N Nl

Nl x

ds eff

Efeff ox

oia oib

oicds clm

Efeff eff

oia oib oic

ker [ log( ) ( )

. ( )]( )

= ++

+ −

+ − + + ++

2

2 8

14

14

2 28

2

14 2

10

2 10

2 10

0 510 2 10

2

2

µ

∆

Flicker Noise


where Vtm is the thermal voltage, µeff is the effective mobiity at the given

bias condition, Leff and Weff are the effective channel length and width,

respectively. The parameter N0 is the charge density at the source given by:

The parameter Nl is the charge density at the drain given by:

∆Lclm refers to channel length reduction due to CLM and is given by:

2. Otherwise,

N0

Cox VGS VTH–( )q----------------------------------------=

Nl

Cox VGS VTH– VDS′–( )q

---------------------------------------------------------=

VDS′ MIN VDS VDSAT,( )=

∆Lclm

Litl

VDS VDSAT–

Litl------------------------------ Em+

ESAT---------------------------------------------

log×

0

=

if VDS > VDSAT

otherwise

ESAT2 Vsat×

ueff---------------------=

Channel Thermal Noise


Where, Slimit is the flicker noise calculated at Vgs=Vth+0.1 and Swi is

given by:

B.5 Channel Thermal Noise

There exists two models for channel thermal noise. Each of these can be toggled

by the noimod flag.



FlickerNoiseSlimit Swi×

Slimit Swi+--------------------------=

SwiN VtI

W L f x

oia ds

eff effEf=

2

364 10

8

3

kTgm gds gmb( )+ +

42

KT

LQ

eff

effinv

µ

Q W L C VA

V vtVinv eff eff ox gsteff

bulk

gsteffdseff= − −

+(

( ))1

2 2



The derivation for this last thermal noise expression is based on the noise

model found in [35].

appendix b: equation list - nikola engineering · · 1999-06-20(δ1=0.001) b.1.2 effective...

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